We present new abstract results on the interrelation between the minimizing movement scheme for gradient flows along a sequence of Γ-converging functionals and the gradient flow motion for the corresponding limit functional, in a general metric space. We are able to allow a relaxed form of minimization in each step of the scheme, and so we present new relaxation results too.

We consider the Cauchy problem for the gradient flow u ′ (t) = −∇φ(u(t)), t ≥ 0; u(0) = u 0 , (⋆) generated by a continuously differentiable function φ : H → R in a Hilbert space H and study the reverse approximation of solutions to (⋆) by the De Giorgi Minimizing Movement approach. We prove that if H has finite dimension and φ is quadratically bounded from below (in particular if φ is Lipschitz) then for every solution u to (⋆) (which may have an infinite number of solutions) there exist perturbations φτ : H → R (τ > 0) converging to φ in the Lipschitz norm such that u can be approximated by the Minimizing Movement scheme generated by the recursive minimization of Φ(τ, U, V ) :We show that the piecewise constant interpolations with time step τ > 0 of all possible selections of solutions (U n τ ) n∈N to (⋆⋆) will converge to u as τ ↓ 0. This result solves a question raised by Ennio De Giorgi in [9].We also show that even if H has infinite dimension the above approximation holds for the distinguished class of minimal solutions to (⋆), that generate all the other solutions to (⋆) by time reparametrization.

The purpose of this paper is to introduce a Minimizing Movement approach to scalar reaction-diffusion equations of the form \partial_t u \ = \ \Lambda\cdot \mathrm{div}[u(\nabla F'(u) + \nabla V)] \ - \ \Sigma\cdot (F'(u) + V) u, \quad \text{ in } (0, +\infty)\times\Omega, with parameters $\Lambda, \Sigma > 0$ and no-flux boundary condition u(\nabla F'(u) + \nabla V)\cdot {\sf n} \ = \ 0, \quad \text{ on } (0, +\infty)\times\partial\Omega, which is built on their gradient-flow-like structure in the space $\mathcal{M}(\bar{\Omega})$ of finite nonnegative Radon measures on $\bar{\Omega}\subset\xR^d$, endowed with the recently introduced Hellinger-Kantorovich distance $\HK_{\Lambda, \Sigma}$. It is proved that, under natural general assumptions on $F: [0, +\infty)\to\xR$ and $V:\bar{\Omega}\to\xR$, the Minimizing Movement scheme \mu_\tau^0:=u_0\mathscr{L}^d \in\mathcal{M}(\bar{\Omega}), \quad \mu_\tau^n \text{ is a minimizer for } \mathcal{E}(\cdot)+\frac{1}{2\tau}\HK_{\Lambda, \Sigma}(\cdot, \mu_\tau^{n-1})^2, \ n\in\xN, for \mathcal{E}: \mathcal{M}(\bar{\Omega}) \to (-\infty, +\infty], \ \mathcal{E}(\mu):= \begin{cases} \int_\Omega{[F(u(x))+V(x)u(x)]\xdif x} &\text{ if } \mu=u\mathscr{L}^d, \\ +\infty &\text{ else}, \end{cases} yields weak solutions to the above equation as the discrete time step size $\tau\downarrow 0$. Moreover, a superdifferentiability property of the Hellinger-Kantorovich distance $\HK_{\Lambda, \Sigma}$, which will play an important role in this context, is established in the general setting of a separable Hilbert space.

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This paper will deal with differentiability properties of the class of Hellinger–Kantorovich distances $$\textsf{HK}_{\Lambda , \Sigma } \ (\Lambda , \Sigma > 0)$$ HK Λ , Σ ( Λ , Σ > 0 ) which was recently introduced on the space $${\mathcal {M}}({\mathbb {R}}^d)$$ M ( R d ) of finite nonnegative Radon measures. The derivatives of $$t\mapsto \textsf{HK}_{\Lambda , \Sigma }(\mu _t, \nu _t)^2,$$ t ↦ HK Λ , Σ ( μ t , ν t ) 2 , for absolutely continuous curves $$(\mu _t)_t, (\nu _t)_t$$ ( μ t ) t , ( ν t ) t in $$({\mathcal {M}}({\mathbb {R}}^d),\textsf{HK}_{\Lambda , \Sigma })$$ ( M ( R d ) , HK Λ , Σ ) , will be computed $${\mathscr {L}}^1$$ L 1 -a.e.. The characterization of absolutely continuous curves in $$({\mathcal {M}}({\mathbb {R}}^d), \textsf{HK}_{\Lambda , \Sigma })$$ ( M ( R d ) , HK Λ , Σ ) will be refined.